1. Integration and Differentiation of Time Histories
Unit 21
Integration and Differentiation of Time Histories

2. Accelerometer
Mechanical vibration is usually characterized in terms of acceleration
The main reason is that acceleration is easier to measure than velocity or displacement
Acceleration can be measured with a piezoelectric, piezoresistive or variable capacitance accelerometer

3. Velocity Criteria
Hunt, Gaberson, Bateman, et al, have published papers showing that dynamic stress is directly proportional to modal velocity (future webinar)
A peak velocity of 50 in/sec is sometimes considered as the shock severity threshold for military components
Allowable building floor vibration limits are typically < 2.0 in/sec
Colin Gordon has established a generic vibration criteria for building floor vibration in terms of velocity (see ISO Generic Vibration Criteria for Vibration- Sensitive Equipment)

4. Velocity Sensor
Velocity measurements require a Doppler laser or a geophone
The laser is expensive and requires a direct line of sight
The geophone is bulky and is intended for seismology measurements

5. Geophone

6. Laser Vibrometer Advantage
No mass loading effect from laser on object.
Disadvantage
Laser system actually measures relative velocity between laser source and object, so laser source must be kept still.
A single point laser vibrometer is used to compare the vibration of two similar guitars

7. Scanning Laser Vibrometer
A Scanning Laser Vibrometer measurement shows the velocity profile of a vibrating turbine blade
The measurement grid has been tailored to match the specific shape of the blade

8. Displacement Sensor
Dynamic displacement can be measured by a linear variable displacement transducer (LVDT)
The frequency response is only suited for low- frequency measurements
LVDTs used to measure traffic-induced vibration on underside of bridge

9. Old School Analog Method for Measuring Velocity & Displacement
Measure vibration with charge mode piezoelectric accelerometer
Analog signal goes through Bruel & Kjaer 2635 signal conditioner
Select acceleration, velocity or displacement output with this knob
Analog integration & double integration applied for velocity & displacement, respectively
Highpass filtering needed to prevent spurious offsets, drifts, etc.
Minimum highpass filtering frequencies:
0.2 Hz for acceleration
1 Hz for velocity & displacement

10. Typical Building Vibration Limits
Transportation Research Board Building Maximum Structure Vibration Criteria
Limiting Peak Particle Velocity
Structure and Condition
(in/sec)
(cm/sec)
Historic buildings,
Certain other old buildings
0.5
~1.3
Residential structures
New residential structures
1.0
~2.5
Industrial buildings
2.0
~5.1
Bridges

11. Hyatt Regency Hotel, Phoenix, Arizona
Typical Elevator Recommended Limits
Parameter
Limit
acceleration/
deceleration
< m/sec^2
Speed
< 7.0 m/sec
Jerk rates
< 2.5 m/sec^3
Sound
< 50 dBa
Ear-pressure change
< 2000 Pa
Fast elevator ride from ground floor to top restaurant!

12. Accelerometer Measurement
Integrated Velocity

13. Hyatt Regency Elevator
Accelerometer Measurement
Differentiated Jerk

14. Integration, Trapezoidal Rule
The integration of a time history is carried out on a “running sum” basis.
Let the acceleration time history be represented by a1, a2, a3, , an.
The velocity time history is calculated as follows.

15. Differentiate, Matlab Function
function[v]=differentiate_function(y,dt) %
ddt=12.*dt;
n=length(y);
v(1)=( -y(3)+4.*y(2)-3.*y(1) )/(2.*dt);
v(2)=( -y(4)+4.*y(3)-3.*y(2) )/(2.*dt);
v(3:(n-2))=(-y(5:n)+8*y(4:(n-1))-8*y(2:(n-3))+y(1:(n-4)))/ddt;
v(n-1)=( +y(n-1)-y(n-3) )/(2.*dt);
v(n) =( +y(n-1)-y(n-2) )/dt;
y = input amplitude
v = output amplitude
dt = time step

16. Sine Example Generate sine function: Amp = 1 Dur = 10 sec Freq = 1 Hz
Sample Rate = 40 Hz
(assume amp unit: G )
Save as: sine_accel

17. Integrate from Acceleration to Velocity
Baseline shift
Mean 61 in/sec
Vibrationdata > Time History > Integrate Input File: sine_accel
Trend Removal = None (prior & after) Output File: sine_vel

18. Integrate from Velocity to Displacement
Ski Slope Effect!
Vibrationdata > Time History > Integrate Input File: sine_vel
Trend Removal = None (prior & after) Output File: sine_disp

19. Integrate from Acceleration to Velocity with Mean Removal
Symmetric Oscillation about zero baseline
Vibrationdata > Time History > Integrate Input File: sine_accel
Trend Removal Prior = None After = Mean Output File: sine_vel

20. Integrate from Velocity to Displacement with Mean Removal
Stable oscillation about zero baseline
But with some distortion
Vibrationdata > Time History > Integrate Input File: sine_vel
Trend Removal Prior = None After = Mean Output File: sine_disp

21. Differentiate from Displacement to Velocity
Vibrationdata > Time History > Differentiate Input File: sine_disp
Output File: sine_vel

22. Review Exercise, Sine Amplitude
Agrees with integration & differentiation results on previous slides
Vibrationdata > Miscellaneous Functions > Steady-state Sine Amplitude

23. Launch Vehicle Separation Test
Filename:
pyro_test.txt

24. Integrate from Acceleration to Velocity
Vibrationdata > Time History > Integrate Input File: pyro_test.txt
Trend Removal = None (prior & after)

25. Integrate from Acceleration to Velocity with HP Filtering
Vibrationdata > Time History > Integrate Input File: pyro_test.txt
Trend Removal Prior: Highpass filter at 30 Hz After: none

26. Recall PSD Synthesis

27. PSD Synthesis Review Generate acceleration white noise
Manipulate the time history via FFTs and inverse FFTs so that its satisfies the PSD specification
Integrate resulting acceleration time history to velocity
Integrate resulting velocity time history to displacement
Remove third-order polynomial trend from displacement
Apply tapering using half-cosine function to beginning and end of displacement
Differentiate displacement to velocity and again to acceleration
Steps 3 through 7 allow the set of acceleration, velocity and displacement time histories to each have zero mean values.